A174586 Number of n X n (0,1) matrices with two 1's in each row having positive permanent.

0, 1, 24, 954, 59040, 5295150, 651354480, 105393619800, 21717404916480, 5554438422838200, 1726882980691176000, 641506478978753110800, 280659563041747649760000, 142843312073975729801785200, 83684308104396267184700784000, 55915646244745131440225950320000

Offset: 1

Author

Vladimir Shevelev, Mar 23 2010

Keywords

• nonn

Comments

a(n) is the normalized volume of the convex hull of (classical) parking functions of length n. - Andrés R. Vindas-Meléndez, Jan 13 2023

References

• Vladimir Shevelev, On the permanent of the stochastic (0,1)-matrices with equal row sums, Izvestia Vuzov of the North-Caucasus region, Nature sciences 1 (1997), 21-38 (in Russian).

Links

• Aruzhan Amanbayeva and Danielle Wang, The convex hull of parking functions of length n, Enumer. Comb. Appl. 2 (2022), no. 2, Paper No. S2R10, 1.
• Mitsuki Hanada, John Lentfer, and Andrés R. Vindas-Meléndez, Generalized parking function polytopes, arXiv:2212.06885 [math.CO], 2022.

Crossrefs

Cf. A001499, A007107, A082491.

Programs

Mathematica

Table[n!/2^n * Sum[(2*i-1)*(2*i-1)!!*Binomial[n,i]*(2n-1)^(n-i-1),{i,0,n}],{n,1,20}] (* Vaclav Kotesovec, Nov 30 2017 *)

Formula

a(2)=1, for n>=3, a(n) = A001499(n) + Sum_{k=1..n-2} (-1)^(k+1)*k!*(C(n,k))^2*(n-k)^k*a(n-k).

a(n) = n!*((n-1)/2^(n-1))*Sum_{i=0..n-2} (2i+1)!!*C(n-2,i)*(2n-1)^(n-i-2). [corrected by John Lentfer, Oct 05 2022]

For n>=2, a(n) = (n!/2^n)*Sum_{i=0..n} (2i-1)*(2i-1)!!*C(n,i)*(2n-1)^(n-i-1).

a(n) = Gamma(3/4)*(sqrt(2)*Pi*e)^(-1/2)*n!*n^(n-1/4)*(1+O(n^((-1/4)+epsilon) with arbitrary small epsilon>0 for sufficiently large n.